Blow-up in F-Theory

In this section we will discuss the F-theory dual of horizontal five-branes [92, 93, 198] as will be essential for our understanding of the five-brane superpotential. As before, letB_Y beP(O_BZ⊕ T) where we denote the associated divisor toT _byT and assume that−Tis an effective divisor ofB_Z. This fibrationp:B_Y →B_Zhas two holomorphic sections denoted byC0andC_∞with

C_∞=C₀−p^∗T. (6.35)

Then, the perturbative gauge groupG=G₁×G₂denoting the group factors from the firstE₈by G1and from the secondE8byG2is realized by seven-branes overC0andC_∞with singularity

90 6. Heterotic/F-theory duality and five-brane superpotential typeG₁andG₂, respectively [173, 175]. On the other hand, components of the discriminant on which∆of vanishing order higher than 1 projecting onto curvesC_{i in}B_Z correspond to heterotic five-branes on the same curves inZ [173, 153, 175]. Consequently, the corresponding seven-branes induce a gauge symmetry of non-perturbative nature due to five-branes on the heterotic side.

For the later application in § 6.3.3 we will consider the enhanced symmetry point with G =E₈×E₈ due to small instantons/five-branes such that the heterotic bundle is trivial. In general, an analysis of the local F-theory geometry near the five-brane curveCis possible [198]

applying the method of stable degeneration [84, 191]. However, since the essential point in the analysis is the trivial heterotic gauge bundle, the results of ref. [198] carry over to our situation immediately.

As follows from the above form ofC_∞(6.35), the canonical bundle of the ruled baseB_Y reads

K_BY = −2C0+p^∗(K_BZ+T)= −C0−C_∞+p^∗K_BZ. (6.36) From this we obtain the classesF,Gand∆of the divisors defined byf,gand∆. To match the heterotic gauge symmetryG=E8×E8, there have to be aI I^∗fibers over the divisorsC0andC_∞ inB_Y. SinceI I^∗fibers requiref,gand∆to vanish to order 4, 5 and 10 overC₀andC_∞, their divisor classes split accordingly with remaining parts

F^′=F−4(C0+C_∞)= −4p^∗K_BZ,

G^′=G−5(C0+C_∞)=C0+C_∞−6p^∗KBZ,

∆^′=∆−10(C0+C_∞)=2C0+2C_∞−12p^∗KBZ.

(6.37)

This generic splitting implies that the component∆^′ can locally be described as a quadratic constraint in a local normal coordinatektoC0orC_∞, respectively. Thus,∆^′can be understood locally as a double cover overC₀respectivelyC_∞branching over each irreducible curveC_i of

∆^′∩C0and∆^′∩C_∞. In fact, near an irreducible curveC_iintersecting, say,C0the splitting (6.37) implies that the sectionsf andgtake the form

f =k⁴f^′, g=k⁵(g₅+kg₆)=k⁵g^′ (6.38)

wheref^′,g₅andg₆are sections ofK_B⁻⁴

Y,K_B⁻⁶

Y ⊗T _andK_B⁻⁶

Y, respectively. The discriminant then takes the form∆=k¹⁰∆^′where∆^′is calculated fromf^′andg^′. Thus, the intersection curve is given by {g₅=0} and the degree of the discriminant∆rises by 2 overC_iwithf^′andg^′vanishing to order 2 and 1. The singular curvesC_{i inY} that occur ing as above are the locations of the small instantons/horizontal five-branes in Z [92, 198]. In the CY fourfoldY the collision of aI I^∗and aI₁singularity overC_iinduces a singularity ofY exceeding Kodaira’s classification of singularities. Thus, it requires a blow-upπ:Be3→BY in the three-dimensional base of the curvesC_i into divisorsD_i. This blow-up is crepant, i.e. it can be performed without violating the CY condition since the shift in the canonical class of the base,K_Be3 =π^∗KBY+Di, can be absorbed into a redefinition of the line bundleL^′₌π^∗L₋D_ientering the Weierstraß equation such that

KY =p^∗(KBY+L_)=p^∗_(KBe

Y+L^′_{)=0. (6.39)}

6.2. Blow-ups and superpotentials 91 To describe this blow-up explicitly, let us restrict to the local neighborhood of one irre-ducible curveC_iof the intersection of∆andC₀. We note that the curveC_iinB_Zis given by the following two constraints

h^′₁=k=0, h^′₂=g5=0 (6.40)

forkandg5being sections of the normal bundleN_C0/BY and ofK_B⁻_Y⁶⊗T, respectively. Then, if Y is given as a hypersurface {P^′=0}, we obtain the blow-up as the complete intersection

P^′=0, Q^′=l₁h^′₂−l₂h₁^′ =0. (6.41)

As in eq. (4.48), we have introduced coordinates {l₁,l₂} parameterizing theP¹fiber. However, at least in a local description, we can introduce a local normal coordinatet toC_{iinBZ such that} g5=t g₅^′ for a sectiong₅^′ which is non-vanishing att=0. Then, by choosing a local coordinate k₁ of theP¹ fiber, we can solve the blow-up relationQ^′ to obtaink =k₁t. This coordinate transformation can be inserted into the constraintP^′=0 ofY to obtain the blown-up fourfold

e

Y as a hypersurface. Thef^′,g^′of this hypersurface are given by

f^′=k⁴₁f, g^′=k₁⁵(g5+k1t g6+ ···). (6.42) In particular, calculating the discriminant∆^′ ofYe, it can be demonstrated that theI1 singu-larity no longer hits theI I^∗singularity overC₀[198]. This way we have one description ofYe as a complete intersection and another as a hypersurface. Both will be of importance for the explicit examples discussed in § 6.3.

To summarize, the F-theory counterpart of a heterotic compactification with full pertur-bative gauge group is given by a CY fourfold withI I^∗fibers over the sectionsC0andC_∞inB_Y. The component∆^′of the discriminant enhances the degree of∆on each intersection curveC_i such that a blow-up inBY is necessary. On the other hand, as previously described in § 2.3, each blow-up corresponds to a small instanton in the heterotic bundle [173, 89], e.g. a hori-zontal five-brane on the curveC_i in the heterotic threefoldZ. Indeed, this can be viewed as a consequence of the observation mentioned above that a vertical component of the discrimi-nant with degree greater than 1 corresponds to a horizontal five-brane [175] as the degree of

∆^′onC₀andC_∞is 2.

Let us now discuss how the moduli maps in eq. (6.28) change during the blow-up proce-dure. To actually perform the blow-up along the curveC_i, it is necessary to first degenerate the constraint ofY such thatY develops a singularity overC_idescribed above. This requires a tuning of the coefficients entering the fourfold constraint, thus restricting the complex struc-ture ofY accordingly loweringh^3,1(Y). Then, we perform the actual blow-up by introducing a new Kähler class associated to the exceptional divisorD_i. Thus, we end up with a new CY fourfoldYe with decreasedh^3,1(Ye) andh^1,1(Be_Y) increased by one. This is also clear from the general argument of ref. [175] that, by enforcing a given gauge groupGin four dimensions, the complex structure moduli have to respect the form of∆dictated by the singularity typeG.

Since the blow-up, dual to the heterotic small instanton/five-brane transition, enhances the gauge symmetryG, the form of the discriminant becomes more restrictive, thus fixing more

92 6. Heterotic/F-theory duality and five-brane superpotential complex structures. In this picture the blow-down can be understood as switching on moduli, decreasing the singularity type of the elliptic fibration.

Similarly, we can understand the moduli map (6.28) from the heterotic side. For each tran-sition between small instanton and five-brane the bundle loses parts of its moduli since the small instanton is on the boundary of the bundle moduli space. Consequently, the indexI reduces accordingly. In the same process, the five-brane in general gains moduli counted by h⁰(C_i,NC_i/Z) contributing to the moduli map.

We close this section by making a more refined and illustrative statement about the mean-ing of the Kähler modulus of the exceptional divisorsD_i in the heterotic theory. To do so we have to consider the heterotic M-theory onZ×S¹/Z2. In this picture the instanton/five-brane transition can be understood as follows: A spacetime-filling five-brane wrappingC_i moves on theS¹/Z2and reaches the end-of-the-world brane where one perturbativeE₈gauge group is located [199]. There, it dissolves into a finite size instanton of the heterotic bundleE. With this in mind the distance of the five-brane on the intervalS¹/Z2away from the end-of-world brane precisely maps to the Kähler modulus of the divisorD_iresolvingC_iinB_Y [198].